Jan O. Kleppe
Mathematics Subject Classification (1991). Primary 14C05; Secondary 14C20, 14B12, 14B15, 14F17
We prove the existence of a generically smooth component V of the Hilbert scheme H(d,g;P4) of smooth connected nondegenerate curves in the projective 4-space for every degree d > 20 and genus g > 3d - 28 (for which H(d,g;P4) is non-empty), and we compute its dimension. Moreover, in the subrange g > 1 + d(d + 3)/14, we expect the component V to be among the "nicest" and smallest components of H(d,g;P4) because its generic curve sits on the "most general" smooth surface allowed. We also prove a corresponding existence result for H(d,g;P5) for every d > 29 and g > 2d - 17, and we get some partial results for H(d,g;Pn), n > 5, as well.
To prove these results we develop a new criterion for a smooth curve, sitting on a smooth surface in Pn, to be unobstructed. Finally we determine a range in the (d,g)-plane where there exist smooth connected curves on a smooth Castelnuovo surface in Pn for n = 4 and 5.